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Difference equations, spline approximations and multiperiod linear programs all give rise to linear equation systems that have a characteristic staircase structure. A staircase system’s variables can be partitioned, into a natural sequence of periods, in such a way that every equation involves at most the variables of two adjacent periods. This paper surveys the numerous properties of staircase systems and of the staircase matrices that comprise their coefficients. Particular attention is given to reduced and sparse staircase forms and to the varieties of Gaussian elimination for solving staircase systems.
Robert Fourer (Sun,) studied this question.